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CHAPTER XI

ELEMENTARY SPECTRAL THEORY

The choice of the subject matter of this chapter has been dictated by two
considerations: (1) it is the first step in one of the main branches of modern
functional analysis, the so-called " spectral theory "; (2) it draws practically on
every preceding chapter for the formulation of its concepts and the proof of
its theorems, and thus may convince the student that the " abstract" develop-
ments of these chapters were not purposeless generalizations.

General spectral theory, being closely linked to the general theory of
integration, falls outside the scope of this first volume, and the reader will
not find any results of that theory in this chapter, with the exception of the
proof of the existence of the spectrum (11.1.3) and a few elementary proper-
ties of the adjoint of an operator (Section 11.5). We have concentrated on the
theory of compact linear operators, which can again be considered as " slight"
perturbations of general operators, although in a sense quite different from
the one which was prevalent in Chapter X; here what is considered as
"negligible" is what happens infinite dimensional subspaces, and the sub-
stance of the main theorem (11.3.3) on compact operators is that when we
add such an operator to the identity, what we get is again a linear homeo-
morphism, provided it is restricted to a suitable subspace of finite codimension.
Compact self-adjoint operators in Hilbert space have a special interest, not
only because it is possible to have much more precise information on their
spectrum than for general compact operators (11.5.7), but also because their
general theory immediately applies to Fredholm integral equations with
hermitian kernel (Section 11.6), and in particular to the classical Sturm-
Liouville problem, which we have chosen as a particularly beautiful illustra-
tion of the power of the methods of functional analysis (Section 11.7).

For more information on spectral theory, and on its powerful applica-
tions, we strongly recommend Courant-Hilbert's classic [10] for its delightful
style and its wealth of information. General spectral theory will be developed

312



	CHAPTER XI ELEMENTARY SPECTRAL THEORY

	The choice of the subject matter of this chapter has been dictated by two considerations: (1) it is the first step in one of the main branches of modern functional analysis, the so-called " spectral theory "; (2) it draws practically on every preceding chapter for the formulation of its concepts and the proof of its theorems, and thus may convince the student that the " abstract" develop- ments of these chapters were not purposeless generalizations. General spectral theory, being closely linked to the general theory of integration, falls outside the scope of this first volume, and the reader will not find any results of that theory in this chapter, with the exception of the proof of the existence of the spectrum (11.1.3) and a few elementary proper- ties of the adjoint of an operator (Section 11.5). We have concentrated on the theory of compact linear operators, which can again be considered as " slight" perturbations of general operators, although in a sense quite different from the one which was prevalent in Chapter X; here what is considered as "negligible" is what happens infinite dimensional subspaces, and the sub- stance of the main theorem (11.3.3) on compact operators is that when we add such an operator to the identity, what we get is again a linear homeo- morphism, provided it is restricted to a suitable subspace of finite codimension. Compact self-adjoint operators in Hilbert space have a special interest, not only because it is possible to have much more precise information on their spectrum than for general compact operators (11.5.7), but also because their general theory immediately applies to Fredholm integral equations with hermitian kernel (Section 11.6), and in particular to the classical Sturm- Liouville problem, which we have chosen as a particularly beautiful illustra- tion of the power of the methods of functional analysis (Section 11.7). For more information on spectral theory, and on its powerful applica- tions, we strongly recommend Courant-Hilbert's classic [10] for its delightful style and its wealth of information. General spectral theory will be developed 312